Angle CAD is an inscribed angle intersecting arc CD.....so..its measure is 1/2 of the arc = 30°
And since angle ABO = 60°, then angle BOA = 180 - 30 - 60 = 90°
So....we have 30 - 60 - 90 right triangle
And the side opposite the right angle = AB = 2BO = 10
And the side opposite the the 60° angle ABO = AO = 5√3 = √75 = the radius of the circle
Draw OB through to intersect the circle at M.....so OM is a radius = 5√3 = √75
So BM = OM - BO = [ 5√3 - 5] = [ √75 - 5 ]
And draw BO through to intersect the circle at N
So.....BN = [ 5√3 + 5] = [ √5 + 5 ]
An
So.....we can use the intersecting chord theorem to find BC
AB * BC = BM * BN
10 * BC = [ √75 - 5 ] * [ √75 + 5 ]
10 * BC = 75 - 25
10 * BC = 50 divide both sides by 10
BC = 50 / 10 = 5
